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10. Hasil dari integral trigonometri adalah [tex]\tt-sin(x)^2cos(x)-\frac{2cos(x)^3}{3}+C[/tex].
Pembahasan
Integral adalah suatu bentuk sistem matematika yang merupakan invers dari sistem turunan dan merupakan batas suatu daerah tertentu.
Rumus Integral Tentu :
[tex]\tt \to\int\limits^b_a {f} \, dx =\int\limits^{b+k}_{a+k} {f} \, dx \\\\\to\int\limits^b_a {k.f(x)} \, dx =k\int\limits^b_a {f(x)} \, dx \\\\\to \int\limits^b_a {f(x)} \, dx =-\int\limits^a_b {f} \, dx \\\\\to\int\limits^a_a {f} \, dx =0\\\\\to\int\limits^b_a {+\int\limits^c_b {f} \, dx } =\int\limits^c_a {f} \, dx[/tex]
Penyelesaian Soal
Nomor 10
Diketahui :
[tex]\tt \int\limits{sin^3(x)} \, dx[/tex]
Ditanya :
Hasil integral trigonometri...?
Jawaban :
[tex]\tt \int\limits{sin^3(x)} \, dx\\\\= \int\limits{sin^2(x)sin(x)} \, dx \\\\Misal: \\u=sin^2(x)\\du=(sin^2(x))\,dx\to gunakan~aturan~rantai:(f(g))'=(f(g))'\times g'\\\\du=2gcos(x)\,dx\\\\du=2sin(x)cos(x)\,dx\\\\\\dv=sin(x)\,dx\\\\\int\limits{dv=\int\limits {sin(x)} \, dx } \\\\v=-cos(x)[/tex]
[tex]\tt sin(x)^2(-cos(x))-\int\limits {-co(x)2sin(x)cos(x)} \, dx \\\\=sin^2(x)(-co(x))+2\int\limits{cos(x)^2sin(x)} \, dx \\\\=sin(x)^2(-cos(x))+2\int\limits {cos(x)^2sin(x)(-\frac{1}{sin(x)})} \, dt\\ \\=sin(x)^2(-cos(x))+2\int\limits{-cos(x)^2} \, dt\to gunakan~integral~tentu:\int\limits^b_a {f(x)} \, dx =-\int\limits^a_b {f} \, dx \\\\=sin(x)^2(-cos(x))-2\int\limits {cos(x)^2} \, dt[/tex]
[tex]\tt =sin(x)^2(-cos(x))-2(\frac{cos(x)^3}{3})\\ \\= -sin(x)^2cos(x)-2(\frac{cos(x)^3}{3})\\ \\= -sin(x)^2cos(x)-\frac{2cos(x)^3}{3}+C[/tex]
Pelajari Lebih Lanjut
Detail Jawaban
Kelas : XII SMA
Kode : 12.2.1
Kategori : Integral Trigonometri
Mapel : Matematika
Kata Kunci : Integral, Integral Trigonometri, Integral Sinus, Integral Biasa